I would check to the original raiser on that flop because he will almost always cbet it. I'd call his cbet seeing as the board is pretty dry and check to him on the turn.

As played, it's fine to continue betting on the ace. Once he calls two streets it's good to check the river; fold to most bets unless he's bluffy because he's not going to bet there with a weaker king or lower pair.

Preflop: Here it's Called CBet. But if the guy just fold to re-steal then you can even consider just 3betting here against him.

As playedFlop: Since he may fold some of his hands what he would CBet I'd prefer just to Check/Call here.

As playedTurn: Yes, likely it's better to bet here again since we get a free showdown very often and he wont bet with weaker hands on river.River: Check/Call against a small bet is a good idea but often I would even consider even Check/Fold unless the beat is really that small.